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Unformatted text preview: Green’s Theorem Theorem 1. Let C be a positively oriented, piecewise smooth, simple closed curve in the plane, and let D be the region enclosed by C . Suppose that P and Q have continuous partial derivatives on an open region containing D . Then Z C P dx + Qdy = ZZ D ( Q x- P y ) dA. Positively oriented means that if you were to walk along the boundary curve, the enclosed region would always be on your left . If C is connected, then this corresponds to tracing the curve counterclockwise. Changing the orientation changes the sign. Piecewise-smooth means that C may consist of a finite number of curves, each of which is smooth, though they need not fit together smoothly. By closed we means the curve starts and ends at the same place, and by simple we mean it doesn’t intersect itself. Green’s theorem is thought of as a generalization of the fundamental theorem of calculus in the following way. The integral of P x- Q y over a 2-dimensional region is given by evaluating P and Q only on the boundary of the region, whereas with usual FTC, an integral of f over an interval (straight line segment) is given by evaluating f at the boundary of the interval (the two endpoints). The proof of Green’s theorem in full generality is a little rough, because one has to make precise the definition of orientation, and deal with possibly some rather strange curves. However the proof in the case where C is anything reasonable is found is most calculus books and is actually quite easy. It involves considering the case where C is a curve given in terms of two functions like in the figures below, and then breaking up the curve into pieces that look like these. Green’s theorem is used to compute line integrals over closed curves in the plane. Lineinto pieces that look like these....
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- Winter '10
- Vector Space, Manifold, Line segment, dy